A four-dimensional \(\mathcal {N}=1\)  theory with an \(R\) -symmetry can be coupled to background new-minimal supergravity [1]}, [2]}. In addition to the metric, new-minimal supergravity [3]}, [4]} comprises the \(R\) -symmetry \(U(1)\)  gauge field and a (globally well-defined) one-form background field.
Consequently, the gauge fields obtained after the circle reduction are the KK photon gauge field \(c\)  arising from the 4d metric, the \(U(1)_R\) -gauge field \(\check{A}\) , and we also have the 3d spin connection \(\omega ^{ab}\)  (we use the \(\check{\,}\)  symbol to distinguish 3d quantities from similar 4d quantities). It is convenient to take \(c=c_i\mathrm {d}x^i\)  to have the dimension of a length and \(\check{A} = \check{A}_i \mathrm {d}x^i\)  to have dimension 0. Similarly, the spin connection has dimension 0 as a one-form.
Up to normalization, we can hence form four independent Chern-Simons terms
\(c\wedge \mathrm {d}c,\qquad \check{A}\wedge \mathrm {d}c,\qquad \check{A}\wedge \mathrm {d}\check{A},\qquad {\rm tr}\Big (\omega \wedge \mathrm {d}\omega + \frac{2}{3}\,\omega \wedge \omega \wedge \omega \Big )~,\) 
